Subgroups — C₈×D₄×C₂₂₃ #28383146

Order 14272.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊C₈×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₂₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
8	C₂×C₈×C₂₂₃	—	2	4	no	—
9	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
11	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₂₃	C₄	1	1	yes	characteristic
13	C₂²⋊C₄×C₂₂₃	C₄	1	2	yes	—
14	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂³×C₂₂₃	C₂×C₄	1	2	yes	—
16	D₄×C₂₂₃	C₈	1	4	yes	direct product factor
17	C₂×C₄×C₂₂₃	—	2	2	no	—
18	C₂×C₄×C₂₂₃	—	2	2	no	—
19	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
23	C₈×C₂₂₃	—	2	2	no	—
24	C₈×C₂₂₃	—	2	1	no	—
25	C₈×C₂₂₃	D₄	1	2	yes	direct product factor
26	C₂²×C₂₂₃	—	2	2	no	—
27	C₂²×C₂₂₃	—	2	2	no	—
28	C₂²×C₂₂₃	C₂×C₈	1	4	yes	complemented
29	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₂₃	—	2	2	no	—
31	C₄×C₂₂₃	—	2	1	no	—
32	C₄×C₂₂₃	C₂×C₈	1	2	yes	complemented
33	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₂₃	#41	1	1	yes	characteristic
35	C₂×C₂₂₃	—	2	4	no	—
36	C₂×C₂₂₃	C₄×D₄	1	1	yes	characteristic
37	C₂×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
38	C₂×C₂₂₃	#131	1	1	yes	characteristic
39	C₂₂₃	C₈×D₄	1	1	yes	characteristic, direct product factor, Sylow
40	C₈×D₄	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×C₈	C₂×C₂₂₃	1	2	yes	complemented
42	C₄×D₄	C₂×C₂₂₃	1	1	yes	characteristic
43	C₄⋊C₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
44	C₂²⋊C₈	C₂×C₂₂₃	1	2	yes	complemented
45	C₄×C₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
46	C₂×D₄	C₄×C₂₂₃	1	1	yes	characteristic
47	C₂²×C₄	C₂²×C₂₂₃	1	2	yes	—
48	C₂×C₈	—	2	4	no	—
49	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
50	C₂×C₈	C₂²×C₂₂₃	1	2	yes	—
51	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
52	C₄⋊C₄	C₄×C₂₂₃	1	1	yes	characteristic
53	C₂²⋊C₄	C₄×C₂₂₃	1	2	yes	—
54	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
55	C₂³	C₂×C₄×C₂₂₃	1	2	yes	—
56	D₄	C₈×C₂₂₃	1	4	yes	direct product factor
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
61	C₂×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
62	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	2	1	no	—
65	C₈	D₄×C₂₂₃	1	2	yes	direct product factor
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂×C₈×C₂₂₃	1	4	yes	complemented
69	C₂²	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	C₂×C₈×C₂₂₃	1	2	yes	complemented
73	C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
74	C₄	#41×C₂₂₃	1	1	yes	characteristic
75	C₂	—	2	4	no	—
76	C₂	C₄×D₄×C₂₂₃	1	1	yes	characteristic
77	C₂	C₂²×C₈×C₂₂₃	1	1	yes	characteristic
78	C₂	#131×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₂₃	C₂²
commutator	C₂	C₂²×C₈×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂³×C₂₂₃
socle	C₂²×C₂₂₃	C₂²×C₄
prime radical	C₂×D₄×C₂₂₃	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₈×C₂₂₃	C₂	semidirect
3. C₄⋊C₈×C₂₂₃	C₂	semidirect
4. C₂²⋊C₈×C₂₂₃	C₂	semidirect
5. C₄×C₈×C₂₂₃	C₂	semidirect
16. D₄×C₂₂₃	C₈	direct (N × H)
25. C₈×C₂₂₃	D₄	direct (N × H)
28. C₂²×C₂₂₃	C₂×C₈	semidirect
32. C₄×C₂₂₃	C₂×C₈	semidirect
39. C₂₂₃	C₈×D₄	direct (N × H)
40. C₈×D₄	C₂₂₃	direct (N × H)
41. C₂²×C₈	C₂×C₂₂₃	semidirect
43. C₄⋊C₈	C₂×C₂₂₃	semidirect
44. C₂²⋊C₈	C₂×C₂₂₃	semidirect
45. C₄×C₈	C₂×C₂₂₃	semidirect
56. D₄	C₈×C₂₂₃	direct (N × H)
65. C₈	D₄×C₂₂₃	direct (N × H)
68. C₂²	C₂×C₈×C₂₂₃	semidirect
72. C₄	C₂×C₈×C₂₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₈×C₂₂₃	2
C₄×D₄×C₂₂₃	2
C₄⋊C₈×C₂₂₃	2
C₂²⋊C₈×C₂₂₃	2
C₄×C₈×C₂₂₃	2
C₈×D₄	223

Subgroups — C8×D4×C223 #28383146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×D₄×C₂₂₃ #28383146